Boundary layer model for vortex fingers in type-II superconductors

Propagating fingerlike patterns in type-II superconductors are studied through a boundary layer model that takes into account the coupling with the temperature of the sample. By formulating an approach based on an interfacial description for a domain of vortices, we determine the shape-preserving fronts and study the properties and scale of the patterns, such as the fingers’ shape and width. We show that the formation and the characteristics of these instabilities are strictly related to the local overheating of the material and depend on the substrate temperature, in agreement with the experiments and suggestions from linear stability calculations.

Figure 1

(Color online) Comparison of the $\theta \left(s\right)$ and $T\left(s\right)$ fields profiles in the cases with discontinuous and continuous functions $f(j,T)$. The data correspond to the values $v_0=1.0193$, flux creep rate $j_1=0.0043$, $\alpha =3.9$, $\tau =0.1$, $j=0.9648$.

Figure 2

(Color online) Fingers shapes in the case with smooth $f(j,T_i)$ for different values of the coefficient $\alpha$ for a velocity $v_0=1.431$ in our dimensionless variables, $j=0.925$, $\tau ={10}^{-1}$ and $j_1=0.004$. By increasing $\alpha$ the fingers become less wide.

Figure 3

Plot of the width $w$ of the fingers vs the coefficient $\alpha$ for parameter values of Fig. 2.